International Journal of Philosophical Studies, 2013 Vol. 21, No. 2, 163–183, http://dx.doi.org/10.1080/09672559.2012.727011 Intrinsic Explanation and Field’s Dispensabilist Strategy Russell Marcus
Abstract Hartry Field defended the importance of his nominalist reformulation of Newtonian Gravitational Theory, as a response to the indispensability argument, on the basis of a general principle of intrinsic explanation. In this paper, I argue that this principle is not sufficiently defensible, and can not do the work for which Field uses it. I argue first that the model for Field’s reformulation, Hilbert’s axiomatization of Euclidean geometry, can be understood without appealing to the principle. Second, I argue that our desires to unify our theories and explanations undermines Field’s principle. Third, the claim that extrinsic theories seem like magic is, in this case, really just a demand for an account of the applications of mathematics in science. Finally, even if we were to accept the principle, it would not favor the fictionalism that motivates Field’s argument, since the indispensabilist’s mathematical objects are actually intrinsic to scientific theory. Keywords: philosophy of mathematics; indispensability argument; Hartry Field; intrinsic explanation
1. Overview Quine argued that we should believe that mathematical objects exist because of their indispensable uses in scientific theory. Hartry Field rejects Quine’s argument, arguing that we can reformulate science with- out referring to mathematical objects. Field provided a precedental reformulation of Newtonian Gravitational Theory (NGT) which has been refined, improved, and extended in the years since his original monograph. In this paper, I argue that Field’s impressive construction and its extension do not impugn Quine’s argument in the way that Field alleges that they do. I do not defend the indispensability argument. I merely attempt to undermine Field’s influential line of criticism. Field’s reformulation of NGT is simply a formal construction. Field argues for its relevance in a defense of nominalism on the basis of a principle of intrinsic explanation. I argue that this principle is not suffi- ciently motivated or defensible, and that it can not do the work for which Field uses it. I start with the relevant background, in x2, and a
Ó 2013 Taylor & Francis INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES discussion of Field’s principle, in x3. In x4–x6, I present and reject Field’s arguments for that principle. In x7, I show that even accepting Field’s principle would not lead to his nominalist, or fictionalist, conclusion.
2. Quine’s Indispensability Argument and the Dispensabilist Response Quine never presented a detailed indispensability argument, though he alluded to one in many places. I interpret Quine’s argument, QI, as fol- lows.
QI1. We should believe the single, holistic theory which best accounts for our sense experience. QI2. If we believe a theory, we must believe in its ontological com- mitments. QI3. The ontological commitments of any theory are the objects over which that theory first-order quantifies. QI4. The theory which best accounts for our sense experience first- order quantifies over mathematical objects. QIC. We should believe that mathematical objects exist.1
An instrumentalist who believes that our uses of mathematics in sci- ence do not commit us to the existence of mathematical objects, may deny either QI1 or QI2, or both.2 Regarding QI1, there is some debate over whether we should believe our best theories. Regarding QI2, one might interpret some of a theory’s references fictionally. I return briefly to instrumentalist responses to QI in x7 of this paper. Quine’s procedure for determining the ontological commitments of theories, QI3, is less controversial than QI1–2, and may be taken as defi- nitional. Still, instrumentalists who dismiss QI should be prepared to defend alternative criteria for determining their ontological commit- ments. One alternative to QI3 would be to adopt an eleatic principle on which the ontological commitments of any theory are, approximately, those objects in the causal realm.3 Debate over QI has focused mostly on QI4. To oppose QI4, Field pro- vided two synthetic reformulations of NGT, replacing the standard ana- lytic version of the theory, which relies on real numbers and their relations, with theories based on physical geometry. A second-order reformulation replaced quantification over mathematical objects with quantification over space-time points. A first-order reformulation referred instead to space-time regions. There are technical questions about whether Field’s reformulations are adequate for NGT. Field mostly ceded the second-order reformulation, due to problems involving
164 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY incompleteness.4 The first-order version, using Quine’s canonical language, is a more appropriate response to QI anyway. There are also questions about whether analogous strategies are available for other cur- rent and future theories.5 I put these questions aside, for this paper, and suppose that reformulations in the spirit of Field’s construction are avail- able for our best theories. My concern in this paper is whether such reformulations are better theories than standard ones, for the purposes of QI. The superiority of dispensabilist reformulations is important because the indispensability argument relies on the claim, at QI1–2, that we find our ontological com- mitments in our best theory. Field defends his reformulation on the basis of a principle of intrinsic explanation. I argue that this principle is false, and that the standard theory is preferable to its dispensabilist counter- part. Thus, I reject Field’s claim that QI4 is false, not because reference to mathematical objects is ineliminable from science, but because the reformulated theories which eliminate quantification over mathematical objects are not our best theories. The value of dispensabilist reformulations has been questioned before. Pincock (2007) argues that the standard theory is better confirmed. To construct representation theorems which demonstrate that a reformula- tion is adequate, the dispensabilist adopts axioms about the physical world and its properties. For example, to measure mass or temperature, Field assumes the existence of spatio-temporal regions or points, and orderings among them, to do the work that connected sets of real num- bers do in the standard theory.6 But, Pincock argues, those assumptions about the physical world are not as well confirmed as the corresponding mathematical axioms and mappings between physical and mathematical structures. Against Pincock’s claim, even if the dispensabilist’s axioms are less well confirmed than the mathematical axioms they replace, they may derive some measure of confirmation from their adequacy. Furthermore, the dispensabilist reformulation, eschewing mathematical objects, makes fewer commitments. It is not clear how to balance the virtue of having fewer commitments with the benefit of having a greater degree of confir- mation. Burgess and Rosen (1997) argue that a better theory should be pub- lishable in scientific journals, and adopted by working scientists; since dispensabilist reformulations are not preferred by practicing scientists, they are no better. This is a wrong way to measure the value of a theory. The practicing scientist wants a useful theory to produce and replicate empirical results. The scientist is mainly unconcerned with ontological commitments. While scientists do seek parsimony among the concrete elements of their theories, an alternative formulation of a given theory whose only advantage is the removal of abstract objects is unlikely to be 165 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES of interest to the readers of a scientific journal. Field’s defense of his reformulation correctly emphasizes concerns about ontological commit- ments. Field shows that it is reasonable, even preferable, to continue using our standard theories, even if we do not really believe that the mathematical objects to which they refer exist. Dismissing concerns about the ontological commitments of our theories, as Burgess and Rosen do, ignores the questions raised by QI about whether scientific theories must quantify over mathematical objects.7 Much of the debate over whether dispensabilist reformulations are better than their standard counterparts has focused on their attractive- ness. Field uses attractiveness as a criterion for acceptable reformula- tions in his original work.8 But, attractiveness is a vague and malleable criterion. One might find a theory attractive based on its strength, simplicity, or explanatory power. It is unclear how to balance such considerations. ‘Of course, it is a deep and difficult question how the various attributes that contribute towards a theory’s attractiveness ought to be spelled out, and how these attributes are to be independently measured and weighed against each other’ (Melia 2000: p. 472). Mark Colyvan, arguing that the standard theory is more attractive than Field’s reformulation, mentions the unification achieved by the standard theory, and its boldness, simplicity, and predictive powers.9 I believe that Colyvan’s argument can be made more precise. This paper pursues and extends Colyvan’s argument, criticizing Field’s own criterion for attractiveness, his principle of intrinsic explanation. I present a spe- cific explanation of why Field’s reformulation is not a better or more attractive theory than the standard one.
3. Intrinsic and Extrinsic Explanations and Theories Field defends his reformulation by appealing to a general preference for intrinsic explanations over extrinsic ones.
If in explaining the behaviour of a physical system, one formulates one’s explanation in terms of relations between physical things and numbers, then the explanation is what I would call an extrinsic one. It is extrinsic because the role of the numbers is simply to serve as labels for some of the features of the physical system: there is no pretence that the properties of the numbers influence the physical system whose behaviour is being explained. (The explanation would be equally extrinsic if it referred to non-mathematical enti- ties that served merely as labels...) (Field 1989b: pp. 192–3)
166 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY Field uses ‘intrinsic’ and ‘extrinsic’ to apply to entities, theories, and explanations. The application to entities is basic, and he classifies explanations and theories depending on the types of objects used. Expla- nations and theories are intrinsic if they make no demand for extrinsic objects. According to Field, numbers are extrinsic to physics, while physical objects and space-time regions are intrinsic.10 Field also applies the intrinsic/extrinsic distinction within mathematics. Numbers are extrinsic to geometry, while line segments and their properties are intrinsic.
The fact that geometric laws, when formulated in terms of distance, are invariant under multiplication of all distances by a positive con- stant, but are not invariant under any other transformation of scale, receives a satisfying explanation [in Hilbert’s geometry]; it is explained by the intrinsic facts about physical space, i.e. by the facts about physical space which are laid down without reference to numbers in Hilbert’s axioms. (Field 1980: p. 27)
The application of ‘intrinsic’ within mathematics proper raises several questions about the relationships among mathematical theories. Are real numbers intrinsic or extrinsic to the theory of natural numbers? Are sets extrinsic to category theory? Are topological spaces extrinsic to Euclidean geometry? Similar questions can be asked purely within empirical science. Are biological or psychological predicates extrinsic to physics? The objects of physics could be considered extrinsic to the special sciences, especially if there are emergent properties in those sciences. In fact, the classification of objects or theories as intrinsic or extrinsic seems suspiciously flexible. Consider how an Aristotelian would deem terrestrial objects as objects extrinsic to theories about planets and stars. These questions about Field’s principle within either mathematics or empirical science should make us wary of the commonsense intrinsic/ extrinsic distinction. But, since my goal at this point is just to illustrate Field’s distinction, and since my concern in this paper is with the rela- tionship of mathematics to physical theories, I shall put them aside. Field presumes that mathematical objects do not influence physical systems to classify them as extrinsic to physical theory.
If, as at first blush appears to be the case, we need to invoke some real numbers … in our explanation of why the moon follows the path that it does, it isn’t because we think that the real number plays a role as a cause of the moon’s moving that way… (Field 1980: p. 43)
167 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES Field’s preference for intrinsic explanations is a broad methodological principle.
Extrinsic explanations are often quite useful. But it seems to me that whenever one has an extrinsic explanation, one wants an intrinsic explanation that underlies it; one wants to be able to explain the behavior of the physical system in terms of the intrinsic features of that system, without invoking extrinsic entities (whether mathematical or non-mathematical) whose properties are irrelevant to the behavior of the system being explained. If one cannot do this, then it seems rather like magic that the extrinsic explanation works. (Field 1989b: p. 193; see also Field 1980: p. 44 and Field 1989a: pp 18–9)
Call this principle PIE: we should prefer intrinsic explanations over extrinsic ones, when they are available. PIE is supposed to account for Field’s preference for synthetic physical theories over analytic ones. PIE also supports Field’s argument for substantivalist space-time (since rela- tionalist theories require references to extrinsic real numbers) and explains his hostility to modal reformulations of science (since modal properties are extrinsic to physics). Field’s focus on intrinsic explanations, rather than intrinsic theories, might seem a bit puzzling. His project is clearly a response to Quine’s indispensability argument, which is formulated in terms of theories because of Quine’s demand that we find our ontological commitments in our best theory. To reformulate the indispensability argument in terms of explanations would force the indispensabilist to argue that we deter- mine our commitments by consulting our best explanations. Though recent work by Colyvan and Alan Baker develops an explanatory indis- pensability argument, that argument is not Quine’s argument, nor is it the argument to which Field is responding. I shall not pursue it here.11 If one thinks that scientific explanations are exhausted by the applica- tions of our most austere scientific theories to sets of initial conditions, then there is no significant difference between appeals to explanations and theories. For traditional covering-law analyses of explanation, we need not worry whether Field’s principle is made in terms of explana- tions or theories. Indeed, Field seems to have such a model of explana- tion in mind.
What we must do is make a bet on how best to achieve a satisfac- tory overall view of the place of mathematics in the world … My tentative bet is that we would do better to try to show that the explanatory role of mathematical entities is not what is superficially appears to be; and the most convincing way to do that would be to 168 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY show that there are some fairly general strategies that can be employed to purge theories of all reference to mathematical enti- ties. (Field 1989a: p. 18; see also fn15)
Furthermore, Field says that an explanation is, ‘A relatively simple non-ad hoc body of principles from which [the phenomena] follow’ (Field 1989a: p. 15). In contrast, one might believe that criteria for good explanations are different from criteria for good theories, especially when theories are used for revealing ontological commitments. One might, say, wish that explanations be perspicuous. If so, one could not prefer Field’s reformu- lation of NGT to the standard theory. Indeed Field’s reformulation is imperspicuous, and hardly recognizable as NGT. It would be impossible to use, which is why he attempts to establish that mathematical theories are conservative over nominalist physical ones.12 If we were to adjust QI to focus on explanation in this sense, a preference for intrinsic explana- tions could not support a dispensabilist reformulation. I will focus on PIE in the sense that I believe Field intended. In this standard sense, explanatory power is an important theoretical virtue.
4. Field’s Motivation for PIE: Hilbert’s Intrinsic Geometry I discern three arguments for PIE in Field’s work. There is the ‘magic’ argument mentioned in the previous section, which I assess in x6. Field also relies on an implicit Okhamist argument, which I consider in x5. Lastly, Field argues that the importance of Hilbert’s 1899 reformulation of Euclidean geometry, which inspired Field’s project, can be explained by PIE: Hilbert’s axiomatization is superior because it is an intrinsic the- ory.13 In this section, I argue that we can understand the success of Hil- bert’s axiomatization without adopting PIE as a general principle which supports Field’s reformulation as a best theory for the purposes of QI. The projects of axiomatizing mathematics in the late nineteenth cen- tury were motivated by diverse factors, two of which stand out: the oddi- ties of transfinite set theory, and the development of non-Euclidean geometries.14 In both cases, traditional mathematical ontology was con- tentiously extended without obvious inconsistency. Rigor in the form of axiomatic foundations was sought to put the controversial new theories on firm ground. Since the development of analysis in the seventeenth century, formula- tions of Euclidean geometry had used real numbers to represent lengths of line segments and triples of real numbers to represent points. Hilbert’s new axiomatization referred to regions of geometric space in lieu of real numbers, and used the geometric properties of betweenness,
169 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES segment congruence, and angle congruence in the way that real numbers, and their ordering, were used in analytic versions. Hilbert constructed representation and uniqueness theorems which assured the adequacy of his so-called synthetic theory. We can understand why Hilbert would prefer a synthetic geometry over analytic versions without appealing to PIE. Here is what Hilbert says about his motivation:
I wanted to make it possible to understand those geometrical prop- ositions that I regard as the most important results of geometric inquiries: that the parallel axiom is not a consequence of the other axioms, and similarly Archimedes’ axiom, etc. I wanted to answer the question whether it is possible to prove the proposition that in two identical rectangles with an identical base line the sides must also be identical, or whether as in Euclid this proposition is a new postulate. I wanted to make it possible to understand and answer such questions as why the sum of the angles in a triangle is equal to two right angles and how this fact is connected with the parallel axiom … (Frege 1980: pp. 38–9)
Hilbert’s account of his motivations make it clear that he wanted to clarify relations within geometry. By relying on geometric relationships to explain geometric phenomena, he avoided worries about the consis- tency of analysis in addition to the worries about geometry. Hilbert thought that his axiomatization better explained geometric entailments. We can best interpret Hilbert’s motivation as purely mathematical, rather than ontological. He makes no suggestion that his new theory is better for the purposes of revealing ontology, which is how Field uses his formulation of NGT. Hilbert devised his axiomatization well before Hempel’s work on scientific explanation, which linked explanation with formal theories, and well before Quine’s work on ontological commit- ment, which linked formal theories with commitments. Furthermore, there are no benefits of parsimony arising from Hilbert’s work. He only shows that real numbers are avoidable in the axiomatization of geome- try. His project was not intended to eliminate commitments to numbers. Regarding more general ontological questions, our main worry within mathematics is antinomy, not parsimony. Worries about antinomy within analysis motivated the arithmetization project of Cauchy, Weierstrass, Dedekind, and others, as well as Hilbert’s axiomatization of geometry. But, contradictions may be more easily discovered in larger, more com- prehensive theories than in smaller, more isolated ones. The superiority of Hilbert’s axiomatization for the purposes of revealing geometric rela- tions does not entail its superiority in constructing proofs and discover- ing contradictions. For a recently well-worn example, consider that 170 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY Fermat’s theorem, a number-theoretic claim, was proved by mapping formulas to topological spaces, after hundreds of years of more direct, intrinsic attempts to prove it. The notion of intrinsic explanation within mathematics proper is clo- sely related to what is called purity. ‘A pure proof or solution is one which uses only such means as are in some sense intrinsic to (a proper understanding of) a theorem proved or a problem solved’ (Detlefsen and Arena 2011: p. 1). Many mathematicians prefer pure proofs, in this sense. But impure, extrinsic proofs are not universally denigrated. Det- lefsen and Arana describe a controversy between Descartes and Wallis, on one side, and Newton and MacLaurin, on the other, concerning the uses of analytic (algebraic) methods in geometry. Pure, intrinsic methods did not prevail. ‘Despite Newton’s reservations concerning algebraic methods, mathematicians of the eighteenth century and later generally followed Descartes and Wallis in sanctioning the relatively free use of “impure” algebraic methods in geometry’ (Detlefsen and Arena 2011: p. 4). Purity, or intrinsicness, is a mathematical value, but one among many. Different axiomatizations serve different purposes. For ontological pur- poses, we are interested in the conjunction of Hilbert’s construction with analysis, which maps geometric structures onto those of number theory. We need not invoke a general principle, PIE, to explain the utility of Hilbert’s reformulation, and insist that his intrinsic, synthetic theory is superior to the analytic formulation. While Hilbert’s axioms emphasize narrow geometric relations, they omit broader, edifying relations between analysis and geometry.
5. Unification and Parsimony Since we need not appeal to PIE to see the virtues of Hilbert’s reformu- lation, Field’s response to QI loses some of its motivation. More impor- tantly, PIE, which seeks to isolate theories and explanations according to their intrinsic elements, conflicts with our general preference to unify theories, revealing connections among diverse disciplines. A comprehen- sive theory simplifies, by showing how different commitments cohere. As Michael Devitt argues, ontology is not to be found in isolated theories. ‘The best ontology will be that of the best unified science’ (Devitt 1984: x4.9; see also x7.8). For examples of the virtues of extrinsic theories within mathematics, consider how the fundamental theorem of calculus bridges geometry and algebra. Algebra could be seen as extrinsic to geometry, but uniting the two theories yields a more comprehensive, and more fruitful, theory.
171 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES Also, we can prove more in an extrinsic second-order theory than we can in an incomplete first-order theory, like first-order arithmetic, itself. The many precedents for unification in science include Newtonian Gravitational Theory unifying Kepler’s celestial mechanics with Galileo’s terrestrial mechanics, and Maxwell’s electrodynamics unifying electrical and magnetic theories with optics. Consider how welcome bridge laws between physics and chemistry or biology would be. Indeed, unification may be central to our notions of scientific explanation. All of the most promising accounts of scientific explanation emphasize unification. Cov- ering laws are preferred if they are broader. Causal accounts seek fewer, more unifying causes. On Kitcher’s unificationist account, it is essential to an explanation that it unify a range of disparate phenomena. Kitcher even argues that unification is the underlying principle which the cover- ing-law model was intended to capture.15 Of course, many good explanations appeal only to isolated portions of our theories. For example, even if we presume that mental-state predi- cates are somehow reducible to physical ones, a psychological explana- tion may not need to appeal to any physical principles. But, such limited explanations do not conflict with our broader presumption toward unifi- cation. Even a dualist would have to appreciate the development of bridge laws between cognitive and physical sciences. It would be implau- sible for Field to reject the general desire for unification, and there is no evidence that he does. But granted that we seek unifying explanations and theories, Field’s general principle of intrinsic explanation seems dif- ficult to defend. One might wonder if the unification of mathematics and physics is a special case which resists a general preference for unification. Such resis- tance might be supported, say, by the observation that mathematical objects are causally independent of the physical world. Does our prefer- ence for unification prevail? Or, do the differences between mathematics and physical science entail that we should prefer scientific theories which eschew extrinsic mathematical objects? In the latter case, our preference for intrinsically isolating physics from mathematics is a specific case, not a corollary of a general principle. Even if one agreed with Field that his reformulation of NGT were a preferable, more attractive theory, that preference would not derive from a general principle of intrinsic explanation. In the former case, we are left to wonder whether a limited principle of intrinsic explanation, in this particular case, supports a preference for Field’s reformulation. Colyvan directly tackled the question of whether Field’s reformulation is more attractive than standard NGT. He argued that unifying mathematics and science leads to a preferable theory. ‘Mathematics contributes to the unification and boldness of the physical
172 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY theory in question, and therefore is supported by well-recognised princi- ples of scientific theory choice’ (Colyvan 2001: p. 81). Colyvan provides three examples. First, the introduction of complex numbers as missing solutions to quadratic differential equations simplifies mathematics, since we need not wonder why some quadratic equations have only one, or even no, root. It unifies exponential and trigonometric functions, and any scientific theory which uses such func- tions. Second, Dirac predicted the existence of positrons by relying on the mathematical solutions to his eponymous equation in relativistic quantum mechanics; positrons were not experimentally verified for another five years. The unification of mathematics and physics allowed for faster scientific progress. Lastly, the Lorentz transformations were initially derived as an account of the failure of the Michelson-Morley experiment intended to provide evidence of the ether. Lorentz, who was in the grip of a false scientific theory, nevertheless developed equations which were later derived from a better scientific theory, viz. special rela- tivity. Without the underlying extrinsic mathematics, it is difficult to see how Lorentz could have developed his equations. Colyvan’s examples illustrate how, in the absence of an over-riding principle, being intrinsic is just one among many characteristics to be weighed when evaluating the attractiveness of a theory. The principle of intrinsic explanation seems especially disfavored when applied specifi- cally to the mathematics used in science, i.e. in the specific case on which QI depends. The unification of mathematics with physics yields a simpler and more powerful theory, a point which Field grants by arguing for conservativeness. And, the isolation of scientific theory from mathemat- ics, especially on the basis of a dispensabilist reformulation, denies important relations among mathematical and physical objects. For a sim- ple example, it is a mathematical property of a three-membered set that it has exactly three two-membered subsets. Applying this property, we can account for why we can, with a red marble, a blue marble and a green marble, form exactly three different-looking pairs of marbles. The ability of a theory to unify disparate phenomena is only one fac- tor among several that we use to evaluate theories. Others include strength, simplicity, fruitfulness, perspicuity, and parsimony. Complete lists are difficult to formulate. Field’s argument is that we should include on such a list whether a theory is intrinsic, and I have argued that our desire to unify theories is more important. Still, the specific case at hand, the one for which Field invokes PIE, is whether standard (extrinsic) sci- ence is preferable to Field’s nominalized (intrinsic) theory. The only obvious theoretical virtue of the nominalized theory is parsimony.16 Desire for parsimony proceeds from a principle applicable to the con- crete objects posited by our best theories: do not multiply physical enti- ties without good reason. When constructing scientific theories, it is 173 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES important not to posit more in the world than that which accounts for the phenomena. It is an open question whether principles of parsimony should apply to the mathematical objects used in standard formulations of scientific theories. In mathematics proper, parsimony is not the most important theoretical virtue. In contrast to the natural scientist, the mathematician explores her/his universe with a desire to multiply entities. In mathemat- ics, it is often a virtue to be plenitudinous, as long as we avoid antimony. Once we have admitted abstracta into our ontology, we do not run out of room. Maddy makes this point especially in regard to set theory.
If mathematics is to be allowed to expand freely in this way, and if set theory is to play the hoped-for foundational rule, then set the- ory should not impose any limitations of its own: the set theoretic arena in which mathematics is to be modelled should be as gener- ous as possible; the set theoretic axioms from which mathematical theorems are to be proved should be as powerful and fruitful as possible. Thus, the goal of founding mathematics without encum- bering it generates the methodological admonition to MAXIMIZE. (Maddy 1997: pp. 210–11)
Set theorists proudly present discoveries of distinct new cardinals. For another example, Kripke models for modal logic have ameliorated mathematical worries about modality without resolving persistent philo- sophical worries about possible worlds. The belief that principles of par- simony are applied differently in mathematics is also a basis for Mark Balaguer’s plenitudinous platonism on which every consistent set of mathematical axioms truly describes a mathematical universe. Worries about the introduction of new mathematical entities, as with complex numbers, or transfinites, tend to focus mainly on their consistency, or the rigor with which they are introduced. PIE would reduce the ontology of scientific theories at the expense of perspicuity, explanatory power, fruitfulness, and coherence with other theories. And it is not even clear that the reduced ontology is preferable. Burgess and Rosen argue against reduced mathematical ontology as a theoretic virtue. ‘It is at least very difficult to find any unequivocal his- torical or other evidence of the importance of economy of abstract ontology as a scientific standard for the evaluation of theories’ (Burgess and Rosen 1997: p. 206). The indispensability argument is alluring since it seems to provide a framework on which mathematical nominalists and platonists can agree.17 Like QI, PIE was intended as a non-question-begging approach to the nominalist/platonist debate. But, the old debate remains.
174 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY 6. The ‘Magic’ Argument I have argued that two of Field’s three arguments for PIE (its ability to explain the value of Hilbert’s geometry and a general preference for parsimony over unification) are unsuccessful. Lastly, in defending his general principle, Field granted the utility of extrinsic explanations, but argued that they seem like magic if there is no underlying intrinsic explanation presupposed. Field’s claim is that explanations of physical phenomena should be possible which refer only to entities which are active in producing those phenomena. The obvious defense of the general demand for intrinsic theories comes from linking theories with ontological commitment, as Quine does. We want our theories to refer only to relevant objects in order to avoid errant commitments. But if we have the mathematical commit- ments already, the extrinsic theory involves us in nothing untoward, and simplifies and unifies our theory. We do not want mistakenly to impute causal powers to mathematical objects by using the extrinsic mathemati- cal theory within physics. But, merely noting that mathematical objects are non-spatio-temporal blocks any such confusion. Thus, the strength of the magic argument depends on whether we have a prior commitment to mathematical objects. The nominalist sees the uses of mathematics in science as magical, since s/he denies the existence of mathematical objects. The platonist sees no magic, only a reasonable demand for an account of the application of mathematics. Several accounts of the application of mathematics in science, compat- ible with platonism, have been developed since Field’s original mono- graph. Mark Balaguer (1998) argues that there is nothing magical about the utility of mathematics in physical science, since mathematics provides a theoretical apparatus for all possible physical states of affairs. Pincock (2004) provides an explanation of the applications of mathematics which, though ontologically neutral, is compatible with platonism and so can also undermine Field’s magic argument. Dissent remains among philosophers of mathematics over whether these particular platonist accounts of the applicability of mathematics are successful.18 Anti-platonist philosophers may see the challenge to provide a platonist account of the applications of mathematics to be insuperable. If they are correct, then Field’s magic argument may hold. But, the existence of recent platonist attempts to account for mathemati- cal applications can give the platonist hope. The resolution of this debate is beyond the range of this paper. More importantly, Field’s magic argument is undermined by the very nature of the epistemology supporting QI. Quine’s holism entails that mathematical objects are intrinsic to physical systems. The indispensabilist
175 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES and the dispensabilist are both committed to the intrinsicness of mathe- matics, as I will discuss in the next section.
7. PIE Does Not Favor Fictionalism I have presented considerations which favor extrinsic theories over intrinsic ones, and which undermine PIE, and thus deflect Field’s criti- cism of QI. In addition to Field’s negative argument against QI4, he pre- sents a positive account of mathematics, which he calls fictionalism. According to fictionalism, mathematical existence claims are false, and mathematical conditionals are vacuously true, if true. In this section, I argue that even if we accept PIE, it does not favor fictionalism. The indispensabilist can accept PIE since s/he should deny that mathematical objects are extrinsic to physical theory. It is well-known that the indispensabilist has trouble accepting a dis- tinction between abstract and concrete objects. As Charles Parsons notes,
Although Quine makes some use of very general divisions among objects, such as between ‘abstract’ and ‘concrete’, these divisions do not amount to any division of senses either of the quantifier or the word ‘object’; the latter sort of division would indeed call for a many-sorted quantificational logic rather than the standard one. Moreover, Quine does not distinguish between objects and any more general or different category of ‘entities’ (such as Frege’s functions). (Parsons 1986: p. 377)
Furthermore, Quine himself wonders if such distinctions are sustainable.
[O]dd findings [in quantum mechanics] suggest that the notion of a particle was only a rough conceptual aid, and that nature is better conceived as a distribution of local states over space-time. The points of space-time may be taken as quadruples of numbers, rela- tive to some system of coordinates... We are down to an ontology of pure sets. The state functors remain as irreducibly physical vocabulary, but their arguments and values are pure sets. The onto- logical contrast between mathematics and nature lapses. (Quine 1986: p. 402; see also Quine 1978; 1960: p. 234; Quine, 1974: p. 88; and Quine, 1969: p. 98)
The indispensabilist’s theory is constructed to explain or represent phenomena involving ordinary objects. ‘Bodies are assumed, yes; they
176 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY are the things, first and foremost. Beyond them there is a succession of dwindling analogies’ (Quine 1976: p. 9). As these analogies dwindle, the traditional abstract/concrete distinc- tion blurs, and so does the intrinsic/extrinsic distinction. For Quine, these distinctions must be made within science. But Quine’s preferred theory does not support them. All of the indispensabilist’s objects are posits of the same monolithic theory, made in the same way, for the same pur- poses of explaining our sense experience. There is no basis for discrete differences in type, no basis for either an intrinsic/extrinsic distinction or a related abstract/concrete distinction. Call this facet of indispensabilism ontic blur. Field classified mathematical objects as extrinsic to NGT in part because of their causal isolation from physical ones. Field, though, is clearly thinking of traditional mathematical objects: abstract objects that exist in all possible worlds, and are knowable a priori, for example. Mark Balaguer defends a principle of causal isolation (PCI) governing the traditional separation of mathematical and physical objects. But, PCI is off limits to the indispensabilist. In fact, Balaguer notes that ontic blur is definitive of QI. ‘The Quine-Putnam argument should be construed as an argument not for platonism or the truth of mathematics but, rather, for the falsity of PCI’ (Balaguer 1998: p. 110). Just as the indispensabilist does not countenance traditional abstract objects, discretely distinct from ordinary objects, any dispensabilist must accept ontic blur. For, the dispensabilist accepts the indispensabilist’s terms of debate, including QI1-3, which are the source of the blur. Given blur, Field can not call mathematical elements extrinsic to physical the- ory. The indispensabilist’s mathematical objects are actually intrinsic to the one, holistic best theory. Field tries to establish that the posits of space-time points differ from posits of mathematical objects in order to admit space-time points as intrinsic to physics. He claims that mathematical objects are supposed to be known a priori, while physical space is not (Field 1980: p. 31). But, for the indispensabilist, mathematical objects, like all objects, are known a posteriori. A defense of the apriority of our knowledge of mathemati- cal objects would undermine both the dispensabilist and the indispensa- bilist, making Field’s reformulation moot. An apriorist could argue for mathematical knowledge more forcefully, independently of QI. Field also argues for the difference between the posits of mathemati- cal objects and space-time points on the basis of the richer mathematical ideology (Field 1980: p. 32). But Resnik (1985) develops an impressive amount of mathematics within Field’s space-time, the geometry of which corresponds to second-order analysis. Not only do we get addition and multiplication over the reals and the natural numbers, but we can set up a coordinate system, and define ordered n–tuples. We can even avoid 177 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES the arbitrary choice of points to serve as 0 and 1 by substituting individ- ual variables. It is difficult to see how Field could deny that numbers are intrinsic to physical theories without turning PIE into some version of an eleatic principle, appealing to the causal isolation of mathematical objects. But, if he is presuming an eleatic principle, it is difficult to see why indispens- ability holds any sway. The eleatic can just deny Quine’s argument in favor of a causal criterion for ontological commitment. The eleatic can be an instrumentalist about a theory’s references, and need not reformu- late physical theory to avoid commitments to mathematical objects. Field’s dispensabilist ideology and the indispensabilist’s quasi-mathe- matical ideology both apply to intrinsic objects. The traditional platonist can make the extrinsic/intrinsic distinction. But by definition, the tradi- tional platonist has an independent epistemology for mathematics. The dispensabilist reformulation of standard science does not denigrate our beliefs about mathematical objects if they are independently justified.
8. Conclusion It is difficult to see any value in PIE, as a general principle guiding the- ory choice. Resnik, reviewing Field’s monograph, argues that we can see it at work in economics.
The Expected Utility Theorem, which underwrites the use of utility functions, establishes that if an agent’s preference ordering satisfies certain conditions then it can be represented by a real valued func- tion which is unique up to positive linear transformations. From this it is usually argued that there is no need to presuppose ill understood utilities in accounting for behavior which maximizes expected utility because an account can be given directly in terms of preferences. (Resnik 1983: p. 515)
Resnik says that an intrinsic account, in terms of preferences, is desir- able because utilities are ill understood. But, if they were better under- stood than preferences, then the account would go the other way. If we could order utilities uniquely, while remaining confused about inter- and intra-personal comparisons of preferences, we would seek to explain preferences in terms of utilities. One principle underlying Resnik’s pref- erence is that we should explain things we do not understand in terms of things we do understand. Appropriate Ockhamist principles also guide the avoidance of utilities. It is ironic that Resnik uses an example which employs mathematics to characterize the elements we understand. If
178 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY utilities were as well understood as mathematical theories, then accounts in terms of them would be welcome. PIE is doing no work, here. I have argued that our preference for unification of theories undermines PIE. A proponent of PIE might complain that once we introduce bridge principles which unify two distinct theories, they are no longer extrinsic to each other, and thus that PIE is not in conflict with unification. Before unification, we have separate theories, and explanations of the principles of one theory in terms of principles of the other would be disfavored. After unification, such explanations would be welcome. To be slightly more precise, consider (the conjunction of axioms of) two completely independent theories, T1 and T2. We could take T1 to be biology and T2 to be quantum mechanics; or, we could take T1 to be ZFC and T2 to be general relativity. But, assume that T1 and T2 are indisputably extrinsic to each other. The theory T1 +T2 which merely conjoins two sets of axioms is thus an extrinsic theory. Explanation of phenomena governed by the axioms of T1 in terms of the principles embodied in T2 would be extrinsic explanations. Now, consider a set of mapping principles, M, which bridge T1 and T2. We can see that Field thinks that T1 +T2 + M is also an extrinsic theory by noting that standard (mathematized) physics includes physical axioms, mathematical axioms, and mappings between the two. These mapping principles are precisely at work when we measure the length of a wire in meters, or when we discuss the Hilbert space of an atom. The defender of PIE who wishes to embrace unification claims that T1 +T2 + M is an intrinsic theory, since the bridge principles connect the objects posited by T1 with the objects posited by T2. This approach would save PIE. We could all agree that extrinsic explanations, in the sense of explanations that used T1 +T2 (without M), were magical, and to be disfavored. But, this interpretation of PIE deprives it of all applica- tion. For, on this view there would be no extrinsic explanations. We would never appeal to mathematics in physics, or to quantum mechanics in biology, unless we had bridge principles in hand. Any plausible expla- nation would have to be intrinsic.19 Unification really is opposed to intrinsic explanation. We can appreciate both intrinsic and extrinsic theories. The situation is analogous to the relation between classical mathematicians and intui- tionists, from a classical perspective. The classical mathematician can appreciate the distinction between constructive and non-constructive proofs, without concluding that only constructive proofs tell us what exists. Similarly, we can appreciate the technical acuity of Field’s con- struction without inferring that there are no mathematical objects. Philosophers with nominalist predispositions may see PIE as a com- monsense principle, and so may have neglected to recognize a gap in Field’s argument against QI. There also may be other reasons to reject 179 INTERNATIONAL JOURNAL OF PHILOSOPHICAL STUDIES QI, or merely to prefer a theory which does not quantify over, or otherwise refer to, mathematical objects. But the principle of intrinsic explanations can not do this work.
Hamilton College, USA
Notes 1 See Quine 1939, 1955, 1958, 1960, 1978, 1980a, 1980b, and 1986. For other versions of the indispensability argument, see Putnam 1975 (the success argument); Resnik 1997: x3.3 (the pragmatic indispensability argument); and Mancosu 2008: x3.2 (the explanatory indispensability argument). I focus on Quine’s argument because Field’s response is directed at it. 2 See Carnap 1950. For more recent defenses of instrumentalism in response to QI, see Melia 2000; Azzouni 2004; and Leng 2005a. 3 Contemporary discussions of the eleatic principle trace mainly to David Arm- strong’s work. Armstrong sometimes focuses on causation (see Armstrong 1978b: p. 46), at other times on spatio-temporal location (see Armstrong 1978a: p. 126). Other formulations are found in Oddie 1982: 286; Azzouni 2004: p. 150; and Field 1989a: p. 68. 4 See Shapiro 1983; Field 1989c, 1990. 5 Burgess and Rosen (1997) elegantly collects the slew of reformulation strate- gies published in the wake of Field’s monograph. See especially the construc- tion at xIIA in the spirit of Field’s original work. Most reformulations replace mathematical references with modal ones. 6 See Field 1980, Chapter 7. See Field 1989b for his arguments for a substantiv- alist interpretation of space-time. 7 On Burgess and Rosen’s suggestion: ‘While entertaining as rhetorical flour- ishes, such demands leave a serious explanatory gap...’ (Pincock 2007: p. 255). 8 See Field 1980: pp. viii, 8 and 41. 9 See Colyvan 1999 and Colyvan 2001: x4.3. 10 Joseph Melia argues that space-time points are actually extrinsic to physical theories; see Melia 1998: pp. 65–7. 11 See Mancosu 2008, x3.2, for a formulation of the explanatory argument, and Baker 2005; Colyvan 2001; Colyvan 2010; and Lyon and Colyvan 2007 for defenses of the argument. 12 If a mathematical theory is conservative over a nominalist physical theory, then we can use the mathematics to facilitate derivations in the physical the- ory with assurance that we will not derive any unacceptable empirical conse- quences. The conservativeness of mathematics would assure us that Field’s reformulation need have no consequences for working scientists. 13 See Hilbert 1971. On Hilbert’s influence, see Field 1980, Chapter 3. 14 Hilbert mentions both in a letter to Frege on his motivation for axiomatizing geometry (Frege 1980: Letter IV/4). 15 See Kitcher 1981: p. 508. 16 The nominalist theory may also be used in an account of the applicability of mathematics in empirical science. But, though the platonist can use any nomi- nalist account just as well, so this does not serve to distinguish the nominalist from the platonist. 17 ‘On the one hand, the indispensability argument sides with nominalists in avoiding any presupposition that mathematical statements are intrinsically 180 INTRINSIC EXPLANATION AND FIELD’S DISPENSABILIST STRATEGY privileged. On the other hand, the argument sides with Platonists in taking mathematical statements at face value, as making genuine ontological claims … This evenhandedness is an important strength of the indispensability argu- ment …’ (Baker 2003: p. 50). 18 See Leng 2005b and Yablo 2005. 19 As an anonymous reviewer notes, Field could object that there is a difference between the kinds of bridge principles involved when T1 and T2 are both scientific theories and when one is scientific and one is mathematical: such principles are causal in the former case and acausal in the latter. In such a response, PIE is doing no work. The difference would be based on an eleatic principle. Such a move, then, would be consistent with the claim of this paper that PIE does not support Field’s reformulation of NGT.
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